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Theorem gencbvex2 2991
Description: Restatement of gencbvex 2990 with weaker hypotheses. (Contributed by Jeffrey Hankins, 6-Dec-2006.)
Hypotheses
Ref Expression
gencbvex2.1  |-  A  e. 
_V
gencbvex2.2  |-  ( A  =  y  ->  ( ph 
<->  ps ) )
gencbvex2.3  |-  ( A  =  y  ->  ( ch 
<->  th ) )
gencbvex2.4  |-  ( th 
->  E. x ( ch 
/\  A  =  y ) )
Assertion
Ref Expression
gencbvex2  |-  ( E. x ( ch  /\  ph )  <->  E. y ( th 
/\  ps ) )
Distinct variable groups:    ps, x    ph, y    th, x    ch, y    y, A
Allowed substitution hints:    ph( x)    ps( y)    ch( x)    th( y)    A( x)

Proof of Theorem gencbvex2
StepHypRef Expression
1 gencbvex2.1 . 2  |-  A  e. 
_V
2 gencbvex2.2 . 2  |-  ( A  =  y  ->  ( ph 
<->  ps ) )
3 gencbvex2.3 . 2  |-  ( A  =  y  ->  ( ch 
<->  th ) )
4 gencbvex2.4 . . 3  |-  ( th 
->  E. x ( ch 
/\  A  =  y ) )
53biimpac 473 . . . 4  |-  ( ( ch  /\  A  =  y )  ->  th )
65exlimiv 1644 . . 3  |-  ( E. x ( ch  /\  A  =  y )  ->  th )
74, 6impbii 181 . 2  |-  ( th  <->  E. x ( ch  /\  A  =  y )
)
81, 2, 3, 7gencbvex 2990 1  |-  ( E. x ( ch  /\  ph )  <->  E. y ( th 
/\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950
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